"Rotznase" is used to describe a child or a kid that behaves in a silly or unfriendly way, it is also used by arrogant persons to name people of lower education or social status. In case it is used for a child it can also have a "friendly" color like "meine süße Rotznase" like a father could name his child in a lovely way.

"Triefnase" is usually used to address adults who are not doing a good job. for example, you hire a guy to clean your car and the car looks worse after the cleaning, you could say this "Triefnase" did a bad job.